We now demonstrate the usefulness of FW-smoothing for optimizing GGF objectives, compared to simply using the Frank-Wolfe method of [1]} with a subgradient of the GGF (FW-subgradient). We note that a subgradient of \(g_{{w}}({x})\)  is given by \({w_{\sigma ^{-1}}},\)  where
\(\sigma \in \operatorname{argsort}(-{x})\) . More precisely, FW-subgradient is also equivalent to using subgradients of \(-g_{{w^1}}\)  and \(-g_{{w^2}}\)  in Line 3 of Alg. REF , instead of \(\nabla f^{\beta _t}(P^{(t)}), \)  ignoring the smoothing parameters \(\beta _t\) . FW-subgradient is simpler than FW-smoothing, but it is not guaranteed to converge [2]}. The goal of this section is to assess whether the smoothing is necessary in practice.
